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The ratio test and root test are both convergence tests used to determine if a series converges or diverges. The main difference between the two is the way in which they compare the terms of the series to its limit. The ratio test compares the ratio of consecutive terms to the limit, while the root test compares the nth root of the absolute value of the nth term to the limit.
To use the ratio test, we take the limit as n approaches infinity of the absolute value of the (n+1)th term divided by the nth term. If the resulting limit is less than 1, the series converges. If it is greater than 1, the series diverges. If the limit is exactly 1, the test is inconclusive and another test must be used.
To use the root test, we take the limit as n approaches infinity of the nth root of the absolute value of the nth term. If the resulting limit is less than 1, the series converges. If it is greater than 1, the series diverges. If the limit is exactly 1, the test is inconclusive and another test must be used.
No, the ratio test and root test are only applicable to series with positive terms. They cannot be used on alternating series or series with negative terms.
If the limit is equal to 0, the test is inconclusive and another test must be used to determine the convergence or divergence of the series.